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Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ g(t) = \dfrac{1}{\sqrt{t}} $

$g^{\prime}(t)=\frac{-1}{2 t \sqrt{t}}$Domain of $g(t)$ and $g^{\prime}(t)$ is $x \in(0, \infty)$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 8

The Derivative as a Function

Limits

Derivatives

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this problem Number twenty six of this tour Calculus, Eighth edition, Section two point eight. Find the derivative of the function using the definition of derivative state the domain of the function and the domain of conservative GFT equals one over square root of team and the definition of the derivative is shown here to the right in green. So we're going to go ahead in use this definition. So the derivative is the limit as h and purchase zero of the function evaluated at T plus age. Ah, minus the function itself. I did all by each. Our first step will be to multiply that numerator denominator by the least common denominator, which is the product of screw the team and this girl from the quantity t plus H King. And when we do this three get to the next step where we had we distribute this amount and we're left with a sort of team minus the square root of T plus each. Ahh, lower H Time squared of team Time's screwed of tip LeSage. Now we do not cry. Some writer denominator die The conjugated the numerator squared of t plus squared of the quantity people teach to the top up and to the bottom. Okay, The reason we choose a conjugal is to take advantage of Ah, the difference of squares Simplicity because this term, by its conjugated, we'Ll just be the first part squared minus second part squared eso This court of t squared is team minus the square root of quantity to proceed squared is t bliss each and that's what the numerator resolves to the denominator is age times square of t times Heard of tp was age times this quantity scrotal t plus square root of the quantity T plus h came in the numerator We have a team and we have a minus t those there goes zero and we left of overthinking it h the h and the numerator Cantel castles with each of the denominator And then as we approach zero each of the H terms and the denominator will go to zero, leaving us with negative one divided by squared of tea. I won't be quiet about this word of people zero or this herd of team and the last part here is squared of T plus the square of t play zero, which is this quart of team. So this bracket is two times the square of a team. So the final answer is negative. One over two times team time is the square root of team. And this is our derivative. If g perm of Team Teo, we want identified the domain malfunction and that I mean, it was derivative. Um, the function here, one over square of t. It has restrictions too. The t as because of its placement and the denominator because it's in the denominator, t cannot be zero and then because the tears and the radical it cannot be negative. So ah, the domain of the function. We will say orgy of team The domain is from his Nero, not including general to positive infinity. So So the domain does not include zero, and it is not included. Um, negative numbers instance. We look at the derivative function and it has the same exact repetition or extractions. And we say that the derivative of tea of GF team also shares the same domain zero two hundred on ly positive numbers. That is our final answer

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